Theoretical Economics Letters, 2013, 3, 302-305
Published Online December 2013 (http://www.scirp.org/journal/tel)
http://dx.doi.org/10.4236/tel.[phone]
Open Access TEL
International Factor Mobility and Dynamic Paths
Hiroshi Goto, Yuji Matsuoka
Graduate School of Economics, Kobe University, Kobe, Japan
Received October 25, 2013; revised November 25, 2013; accepted December 2, 2013
Copyright © 2013 Hiroshi Goto, Yuji Matsuoka. This is an open access article distributed under the Creative Commons Attribution
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ABSTRACT
The aim of this paper is to consider the adjustment process of the spatial structure in a two-country economy where both
labor and capital are mobile. For this purpose, we combine the model of New Econo mic Geograph y with the phase dia-
gram technique. We show that the agglomeration processes are not always monotonic since the mobile factors interact
with each other. More specifically, even when both factors are eventually agglomerated to one country, it is possible
that labor and capital move in oppo site directions in the adjustment process. Differences in factor endowment ratio and
market size play significant roles in this transition path.
Keywords: Labor Mobility; Capital Mobility; Spatial Equilibrium; Adjustment Process; Phase Diagram
1. Introduction
This paper examines dynamic paths of agglomeration.
We construct a two-country, two-good, and two-factor
model, and let both factors move between countries gra-
dually. Because the adjustment processes of the two fac-
tors interact with each other, we expect that dynamic
paths of agglomeration would become complex.
Related to our concerns, the New Economic Geogra-
phy (NEG) literature has provided rich theoretical in-
sights on the factor mobility and distribution of economic
activity across the geographical space. However, while it
has mainly focused on sp atial equilibrium, it seldom pre-
sents explicit details on dynamic paths of agglomeration.
This paper aims at filling the gap1.
Our results show that agglomeration processes are not
always monotonic. When the difference in the factor en-
dowment ratio between countries is large, the factors are
adjusted toward the symmetric equilibrium, reducing the
factor endowment differentials. Once the difference be-
comes sufficiently small, both factors are fully agglomer-
ated to one country because of the scale merit. As a result,
dynamic paths of agglomeration acqui re a distort ed shape.
The remainder of this paper is organized as follows.
Section 2 presents the model. In Section 3, we describe
the transition of the spatial structure. Section 4 con-
cludes.
2. The Model
Our model is based on the footloose capital model, which
is established by Martin and Rogers [2]. We consider an
economy with two countries (Home and Foreign), which
are denoted by
H
and
F
, respectively. In this econ-
omy, there are two sectors: one is the homogeneous ag-
ricultural goods sector and the other is the differentiated
manufactured go ods sector.
We assume that there are individuals. Each indi-
vidual owns one unit of labor and on e unit of capital and
decides where to live and invest. The preference of an
individual is represented by the following Cobb-Douglas
utility function:
L
()
1
1
1,
1
UMA
μμ
μ
μ
μμ
−
−
=−
where
μ
is the share of expenditure on manufactured
goods, A is the consumption of agricultural goods, and M
is the quantity index of manufactured goods. The quan-
tity index of manufactured goods takes the following
Dixit and Stiglitz [3] type CES function:
1We can refer to Baldwin and Venables [1] as a related paper. To con-
sider the international factor mobility, they have shed light on roles o
f
distortion and expectation, while the key factors in our model are factor
endowment ratio and market size. See their
p
a
p
er for more details.
()
11
d,Mq
σ
σσ
σ
ωωσ
−−
Ω

=>


1,

H. GOTO, Y. MATSUOKA 303
where is the set of varieties in the manufacturing
sector, is the consumption of variety , and
is the elasticity o f substitution among varieties. Then ,
the well-known price index corresponding to M is written
as follows:
Ω
q
()
ω ω ∈Ω
σ
()
1
11
dPp
σσ
ωω
−−
Ω

=
,


)
(1)
where is the price of variety .
()
p ω
ω
Next, we describe producers’ behavior. In the agricul-
tural sector, labor is the only input and technology pro-
vides constant returns to scale; hence, one unit of labor is
required for one unit of output. We assume that this sec-
tor is perfectly competitive and that agricultural goods
can be traded between countries at zero transportation
cost. If we use this agricultural good as the numeraire, as
long as it is produced in both countries, the wage rate in
both countrie s can be fi xed to unity.
Firms in the manufacturing sector use one unit of capi-
tal as the fixed input and units of labor as the margi-
nal input. We further assume an iceberg-type transporta-
tion cost in the trade of differentiated manufac-
tured goods between c ountries.
a
(
1 τ >
The profit function of each firm located in
country is
ω ∈Ω
()
,ii HF=
()() ()() ()
() ()
,
iiiiiijij
iiij i
pq pq
aq q r
πωω ωω ω
ωτ ω
=+

−+−

where is the price of the variety produced in
country and sold in country , is the quan-
tity of the variety produced in country and sold in
country , and is country
i’s rental rate of capital.
()
ij
p ω
i
j
j
()
ij
q ω
i
i
Each firm sets its price to maximize its profit. Then,
the price becomes2
r
() ()
,,,
11
ii ij
aa
,.
p
pij
στσ
ωω
σσ
===
−−
HF
(2)
With these prices, we can rewrite country i’s price in-
dex, , in the following way:
i
P
()
1
1,,, ,,
1
iij
a
PnnijHFi
σ
σφ
σ
−
=+ =
−j≠
where represents the freeness of trade, which
takes 0 to 1, while i is the number of firms located in
country . Since each firm uses one unit of capital as the
fixed input, equals the amount of capital invested in
country .
()
1 σ
φτ
−
=
i
i
n
i
n
Given the zero profit condition , the rental rate of capi-
tal in country is i
,,
1
i
i
aq
riH
σ
==
−
where iii ij is the total output of the firms lo-
cated in country .
qq q τ =+
i
Since the market must be clear, , is written as
i
q
,,, ,,
jj
ii
iiiij ii
EP
EP
qi
PpPp
σσ
φ
μ

 

=+ =
 

 

jHFij ≠
(3)
where is the total expenditure of country .
i
Using the above results, we can obtain the rental rate
of capital in country in the following way:
Ei
i
,,, ,.
j
i
iijij
E
E
rij
nn nn
φ
μ
σφφ

=+ =

++


HFij
≠
(4)
3. Adjustment Process to Spatial Equilib ria
For convenience, hereafter, we normalize the values of
labor and capital to unity and consider the model ac-
cording to a share basis. We also introduce a new vari-
able,
H
λ , which represents the share of capital invested
in the Home country.
To consider the adjustment process of capital, follow-
ing standard NEG models, we introduce the following
adjustment process:
[
]
()
1,
HKHFH H
ar r λλ =− −

λ
where is the adjustment speed parameter of
capital. With these dynamics and by using Equation (4)
and certain calculations, we can obtain the following
result:
()
0
K
a>
()
21
0 as ,
1
H
H
HH
s
s
φ
λλ
φ
−
+−

(5)
where
H
s
is the share of labor in the Home country.
This result shows the direction of the ad justment of capi-
tal.
Next, we consider the adjustment process of labor.
Because wages are fixed to unity, the real income of a
worker who lives in country and invests in country
becomes i
j
()
1,,,rPijHF
μ
+=
ji . Workers pr efer the
country in which they can earn higher real incomes with
the given investment destinations. Therefore, we con-
sider the fol l owing dyn amics:
()
11 11
1,
HF
HH FF
HL L
HF HF
HH
rr rr
sa a
PP PP
ss
μμ μμ



 
++ ++
=−+−



 



 


−

where is the adjustment speed para-
meter of labor in the Home (Foreign) country. The point
of this adjustmen t process is to show that under the situ a-
tion of not changing where to invest, each worker
chooses his or her residence by comparing levels of real
income. We can thus easily show the direction of the
(
0
HF
LL
aa>>
)
0
, F
2Hereafter, since every firm sets the same price, we omit the variety
index.
Open Access TEL

H. GOTO, Y. MATSUOKA 304
adjustment of labor as follows:
1
0 as .
2
HH
s λ
 (6)
From Equations (5) and (6), we can draw a phase dia-
gram as Figure 13. We can use this adjustment process to
explain two cases. First, let us consider the case that the
initial value of
(
is at point
)
,
HH
s λ
A
in Figure 1.
Because the initial value is not on the saddle path, the
full agglomeration to the Home country occurs in the
long run. At first glance, when we see that the final con-
figuration of the spatial structure is the full agglomera-
tion, we may consider that both labor and capital mono-
tonically move to the Home country. However, this is not
necessarily true. It is also possible that while capital
moves to the Home country, labor moves to the Foreign
country. Since the amount of capital is small relative to
the size of labor under the line , the rental rate of
capital in the Home country is higher than that in the
Foreign country. This leads to the relocation of capital
from the Foreign country to th e Home country.
0
H
λ =

By contrast, under the line , the number of
firms located in the Home country is absolutely small,
and workers living in the Home country change their
residences to the Foreign country. Then the share of la-
bor in the Home country decreases. After crossing line
, both labor and capital increase in the Home
country because while the amount of capital relative to
the size of labor in the Home country is small, the
amount of capital invested in the Home country is abso-
lutely larger than that in the Foreign country. Th en, since
0
H
s=

0
H
s= 
1
1

1
1
2


0


0


1
1


0
1
2
A
B
Figure 1. Phase diagram and dynamic paths.
the location of the Home country is attractive for both
labor and capital, both factors move to the Home coun-
try.
Next, we consider the second case, namely when the
initial value of
(
is at point in Figure 1.
Because the initial value is not on the saddle path, the
full agglomeration equilibrium to the Foreign country
occurs in the long run. As in the first case, the adju stment
process is not monotonic. In the right-hand area of the
line , since the size of labor is abundant relative
to the amount of capital, capital flows into the Home
country. However, because the number of firms located
in the Home country is absolutely smaller than that in the
Foreign country, workers change their residences from
the Home country to the Foreign country. After crossing
line , while the amount of capital is absolutely
small in the Home country, it is large relative to the size
of labor, and thus, both factors move to the Foreign
country.
)
,
HH
s λ
B
0
H
λ =

0
H
λ =

In these ways, both factors change locations according
to the factor endowment differentials and interact with
each other in their adjustment processes. Finally, we re-
fer to the role of transportation cost, which is a key pa-
rameter in the NEG model. Although transportation cost
plays an important role in determining the slope of
, it does not affect the adjustment process qualita-
tively at all.
0
H
λ =

4. Conclusions
In this paper, we presented a phase diagram in order to
show the dynamic pat hs o f agglome ra t ion.
Under the situation that two factors are mobile, we
showed that adjustment processes of these two factors
interact with each other, and thus, they become compli-
cated. Moreover, the adjustment process is not always
monotonic. In particular, we showed that even if eco-
nomic activities are ultimately agglomerated only to one
country, it is possible th at, on this tran sition path , the two
factors change their locations by moving in opposite di-
rections.
Our analysis is tentative, and hence, there are many
remaining issues. We point out only two extensions here.
First, in our model, if the economy is not on the saddle
path, full agglomeration to one country always occurs.
This result may be extreme. To analyze the dispersion of
economic activity, the present model could be extended
to incorporate immobile factors such as land. Second, we
assume a myopic adjustment process for simplicity. If we
consider expectation in the location decision, the adjust-
ment process radically changes (e.g., Baldwin [4]; Krug-
man [5]). We intend to consider how our results might be
modified by introducing such topics in future research.
3Point
()(
,12,1
HH
s λ =
)
2 is a saddle point. Proof on the existence o
f
the saddle path would be provided on request.
Open Access TEL

H. GOTO, Y. MATSUOKA
Open Access TEL
305
5. Acknowledgements
The authors thanks to No ritsugu Nakanishi, No buaki Ha-
maguchi, Fumio Dei, Yasukazu Ichino, Takashi Shibata,
Yang Xi, Chihiro Inaba, Miwa Nakai and participants of
Rokko Forum at Kobe University, the 23th KMSG at
Kushiro Public Un iversity of Economics and the seminar
at Konan University. The second author was a JSPS re-
search fellow and this work was financially supported by
Grant-in-Aid for JSPS Fellows (No. 10J02314). This
work was also supported in part by Grants for Excellent
Graduate Schools, MEXT, Japan. Needless to say, any
errors remaining in this paper are the responsibility of the
authors.
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